Compatibility of discrete and continuous sampling in sampling theory

The sampling theory uses various formulas for calculating the random error in sampling from homogeneous and heterogeneous test arrays. In the case of discrete sampling from homogeneous arrays, the formulas of the fundamental sampling error obtained heuristically and analytically are used. With continuous sampling from heterogeneous arrays, there are no formulas for calculating errors. This led to the conclusion that the theory of testing is incomplete and doubts about its validity. With the introduction of a new concept — sample sampling — it is shown that the formula of the fundamental sampling error describes the sampling of both homogeneous and heterogeneous arrays. Sampling from homogeneous arrays is considered as a special case of sampling from heterogeneous arrays, technically simplifying the procedure for forming a combined sample. Point sampling allows to obtain the smallest possible mass of the combined sample for a given error. When sampling point samples containing more than one piece, the minimum sample mass increases in proportion to the number of pieces and the dispersion of point samples. The heuristic and analytical formulas of the fundamental sampling error coincide in structure, but differ in the validity of the values included in them and the possibilities of measuring them in real sampling conditions. The mathematical theory of sampling is justified by the condition of coincidence of the formulas of random sampling errors for homogeneous and heterogeneous arrays during the sampling of point samples. The compatibility of discrete and continuous sampling in sampling theory is of practical importance, allowing the transition from single-point sampling to mass sampling. This allows you to calculate the mass of point samples and proceed to calculating the number of point samples based on the residual dispersion. Since the amount of residual dispersion can be very large, the need for high-frequency sampling is justified, which ensures the receipt of up to 20,000 point samples from the batch of the tested product.

Kozin V. Z., Komlev A. S. Compatibility of discrete and continuous sampling in sampling theory. MIAB. Mining Inf. Anal. Bull. 2025;(1-1):145-154. [In Russ]. DOI: 10.25018/ 0236_1493_2025_11_0_145.

The study was supported by the Ministry of Science and Higher Education of the Russian Federation No. 0833-2023-0004 in accordance with the state assignment for the Ural State Mining University.

V.Z. Kozin¹, Dr. Sci. (Eng.), Professor, Head of Chair, Dean of the Mining and Mechanics Faculty, e-mail: gmf.dek@ursmu.ru,
A.S. Komlev¹, Cand. Sci. (Eng.), Senior Researcher, e-mail: tails2002@inbox.ru,
¹ Ural State Mining University, 620144, Ekaterinburg, Russia.

A.S. Komlev, e-mail: tails2002@inbox.ru.

1. Gy P. Sampling of particulate material: Theory and practice. Elsevier, Amsterdam, 1982, 431 p.

2. Dihalu D. S., Geelhoed B. A new multi-axial particle shape factor-application to particle sampling. The Analyst. 2011, vol. 136, no. 18, pp. 3783—3788. DOI: 10.1039/c1an15364a.

3. Geelhoed B. Is Gyʼs formula for the Fundamental Sampling Error accurate? Experimental evidence. Minerals Engineering. 2011, vol. 24, no. 2, pp. 169—173. DOI: 10.1016/j.mineng.2010.11.005.

4. Geelhoed B. Approaches in material sampling. Delft University Press, 2010, 152 p.

5. Napier-Munn T. J., Whiten W. J., Faramarzi F. Bias in manual sampling of rock particles. Minerals Engineering. 2020, vol. 153, article 106260. DOI: 10.1016/j.mineng.2020.106260.

6. Francois-Bongarcon D., Gy P. The most common error in applying ʼGyʼs Formulaʼ in the theory of mineral sampling, and the history of the liberation factor. The Journal of the South African Institute of Mining and Metallurgy. 2002, vol. 102, no. 8, pp. 475—479.

7. Lotter N. O., Evans C. L., Engstom K. Sampling — A key tool in modern process mineralogy. Minerals Engineering. 2018, vol. 116, pp. 196—202. DOI: 10.1016/j.mineng.2017.07.013.

8. Kazimirov M. P., Nikitenko E. M., Lukinykh V. E., Novikova T. M. Applicationof technological methodsof samplepreparation for evaluationof large-volumegold ore objects onthe exampleof the Natalka deposit. Zolotodobycha. 2008, no. 119, pp. 16—21. [In Russ].

9. Engströrn K., Esbensen K. H. Evaluation of sampling systems in iron concentrating and pelletizing processes — Quantification of Total Sampling Error (TSE) vs. process variation. Minerals Engineering. 2018, vol. 116, pp. 203—208. DOI: 10.1016/j.mineng.2017.07.008.

10. Ramsey M. H., Ellison S. L. R., Rostron P. Measurement uncertainty arising from Sampling: A guide to methods and approaches. 2 nd edition, Eurachem, 2019, p. 109.

11. Svensmark B. Extensions to the Theory of Sampling 1. The extended Gy's formula, the segregation paradox and the fundamental Sampling uncertainty (FSU). Analytica Chimica Acta. 2021, vol. 1187, article 339127. DOI: 10.1016/j.aca.2021.339127.

12. Sona M., Dube J.-S. Sampling particulate matter for analysis — Controlling uncertainty and bias using the theory of sampling. Analytica Chimica Acta. 2021, vol. 1185, article 338982. DOI: 10.1016/j.aca.2021.338982.

13. Karpenko N. V. Oprobovanie i kontrol' kachestva produktov obogashcheniya rud [Testing and quality control of oredressing products], Moscow, Nedra, 1987, 216 p.

14. Kulikov A. A., Kulikova A. B. Tekhniko-metodicheskie osnovy oprobovaniya gornykh porod na zoloto [Technical and methodological foundations of testing rocks for gold], Moscow, Nauka, 1988, 112 p.

15. Burdonov A. E., Novikov Yu. V., Lukyanov N. D. Application of the regression analysis apparatus for processing the results obtained during ore processing by the centrifugal concentration method. Tsvetnye Metally. 2024, no. 5, pp. 15—22. [In Russ]. DOI: 10.17580/tsm.2024.05.01.

16. Zakharova A. A., Voitekhovsky Yu. L. Statistical forecast of ore enrichment parameters based on observations in grinds. Obogashchenie Rud. 2024, no. 2, pp. 27—31. [In Russ]. DOI: 10.17580/or. 2024.02.05.

17. Blekhman L. I. Regularities of averaging nonlinear dependencies on fluctuating parameters: application to problems of mineral enrichment. Obogashchenie Rud. 2023, no. 5, pp. 22—28. [In Russ]. DOI: 10.17580/or.2023.05.07.

18. Lepilova I. V. Applicability of GOST 14180-80 requirements for sampling large-batch products on conveyorbelts. Nauchnye osnovy i praktika pererabotki rud i tekhnogennogo syr'ya: Materialy XXIX Mezhdunarodnoy nauchno-tekhnicheskoy konferentsii [Scientific foundations and practice of processing ores and man–made rawmaterials: Materials of the XXIX International Scientific and Technical Conference], Ekaterinburg, 2024, pp. 106—112. [In Russ].

19. Usherov A. I., Ishmetyev E. N., Lyapin A. G., Yamshchikov A. V., Tsyganov A. M. Continuous controlof the chemical composition of sulfide copper-zinc ore. Zavodskaya laboratoriya. 2014, no. 4, vol. 80, pp. 69—73. [In Russ].

20. Nurseitov B. N. Ekonomika metallurgicheskogo predpriyatiya [Economics of a metallurgical enterprise], Karaganda, Izd-vo KarGU, 2009, 297 p.